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Full text of "Treatise On Analysis Vol-Ii"

9   METRIZABLE GROUPS      .41

Hence the property of being a Cauchy sequence with respect to some left-
invariant distance is independent of the choice of distance, and depends only
on the topology of G. We shall say simply that such a sequence is a left
Cauchy sequence in G. A right Cauchy sequence in G is defined analogously
by replacing x~*xm in (12.9.2) by xnx~l. It can happen that a left Cauchy
sequence is not a right Cauchy sequence (Problem 8); but if (jcj is a left
Cauchy sequence, then (x"1) is a right Cauchy sequence. Since the mapping
x\-+x~l is continuous it follows that, if every left Cauchy sequence in G
converges, then so does every right Cauchy sequence. In this case G is said to
be a complete metrizable group: it is complete with respect to every left-
invariant distance and every right-invariant distance defining the topology
ofG.

(12.9.3) Let G, G' be two metrizable groups. Then every continuous homo-
morphism f of G into G' is uniformly continuous with respect to left- (resp.
right-) invariant distances on G and G7.

Let d, d' be two such distances on G, G', respectively. By hypothesis, for
each e > 0 there exists d > 0 such that the relation d(e, z) ^ 5 implies
d'(e',f(z))e. Hence if d(x,y) = d(e, x~ly)  6, then d'(f(x\ f(y))  e,
because

d'(f(x),f(y)) = d'(S, (/Mr VO)) = d'(e',f(x-*y)-)
since /is a homomorphism.

(12.9.4) Let Gl5 G2 be two metrizable groups such that G2 is complete, and
let H! (resp. H2) be a dense subgroup ofGt (resp. G2). Then every continuous
homomorphism u : Hj -> H2 can be extended uniquely to a continuous homo-
morphism u : Gi ~~> G2 . If Gx is also complete and if u is an isomorphism (of
topological groups) of Hj onto H2 , then u is an isomorphism (of topological
groups) of G! onto G2 .

There exist left-invariant distances on Gl and G2 , and the existence of the
continuous extension u then follows from (12.9.3) and (3. 15.6). The fact that u
is a group homomorphism follows from the principle of extension of identities
applied to the two functions (jc, y) i-> u(xy) and (x, y)\-+u(x)u(y) on Gl x Gx
(having regard to (3.20.3)). Finally, if Gx is complete and ifv : H2 - Hj is the
inverse of the isomorphism u, then v can be extended to a continuous homo-
morphism v of G2 into Gx . Since v  u and u o v agree with the identity
mappings on Ht and H2 respectively, they are the identity mappings on G! and
G2 respectively, by virtue of the principle of extension of identities and (3.11.5).
This completes the proof. the